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Question:

True/False?

There exists a continuous surjective map from the complex plane onto the non-zero reals.

Hint:

Search for topological invariants.

Discussion:

Under a continuous function, connected set must go to connected set. The complex plane $$\mathbb{C}$$ is connected.

It’s image must be connected.

$$\mathbb{R}-0$$ is not connected.

So the statement is False.